The newspaper said it still intended to provide the FA with "the relevant transcripts", as it had stated it would earlier in the week.
"However, the police have asked to review this information first. The FA and the Premier League are aware of this," the paper added in a statement.
The FA had earlier urged the newspaper to provide "full and unfettered disclosure of all available material" from the investigation.
The newspaper's revelations have so far led to England manager Sam Allardyce and Barnsley assistant manager Tommy Wright losing their jobs.
The long-running investigation by the newspaper also suggested eight current and former Premier League managers had received 'bungs', or illicit payments, for player transfers.
In another twist to the story Pino Pagliara, one of the football agents to make allegations of financial wrongdoing by managers in an interview secretly filmed by journalists posing as businessmen, said on Friday that his claims had been lies.
"They wanted me to tell them that my relationship with managers was such that a manager would take money from me," he told the BBC.
"I felt that if I didn't impress them they would find somebody that would. I allowed them to believe the managers would not drop the money on the floor if I gave it to them."
Southampton assistant manager Eric Black was the latest to be named in the newspaper investigation and he has denied making any suggestion to undercover reporters that football officials should be paid during transfer negotiations.
Stoke City manager Mark Hughes told reporters on Friday that any wrongdoers should be named without delay.
"If there's evidence certain individuals have been involved in things they shouldn't have been, then it's important those guys are named because if they aren't then everybody else is guilty by association by virtue of being a manager or coach in football," he said.
Second tier Queens Park Rangers said they were unable to proceed with an internal investigation into the conduct of manager Jimmy Floyd Hasselbaink because of the lack of detailed information.
Hasselbaink said he was naive to be sucked into the scandal but "taking money is not what I stand for".
"I have never done it, I would never do it just to get a player to the club so I can benefit from that. No. I want to win games," the Dutchman told Sky Sports television.
QPR said in a statement: "We urge The Telegraph to provide full disclosure of all its information relating to the allegations, including video footage and a full transcript of the discussions that took place.
"The club believes this information should not be provided selectively, but unedited and unconditionally, in order for the club to view the full context and the sequence of what was said by all parties to include in its investigation."
The FA said in its statement that it would be meeting with City of London Police next week.
"The FA treats any allegations of this nature seriously and is committed to investigating them thoroughly, in conjunction with any other appropriate body," it added
Allardyce was sacked on Tuesday after the FA said he had behaved inappropriately following secret filming that showed him offering advice to businessmen on how to circumvent rules on player transfers.
(Reporting by Alan Baldwin in London and Ian Rodricks in Bengaluru, Editing by Ed Osmond and Toby Davis)
